In this paper, we give the  $L^2$ estimates for the Marcinkiewicz
integral with rough variable kernels associated to surfaces. As
corollaries of this result, we show that similar properties still
hold for parametric Littlewood-Paley area integral and parametric
$g_\lambda^\ast$ functions with rough variable kernels. We also
show some sharp difference betweeen properties of singular
integrals and the Marcinkiewicz integral with rough variable
kernels. Some of the results are extensions of some known results.